The Dynamical Fine Structure of Iterated Cosine Maps and a Dimension Paradox
نویسندگان
چکیده
We discuss in detail the dynamics of maps z 7→ aez + be −z for which both critical orbits are strictly preperiodic. The points which converge to ∞ under iteration contain a set R consisting of uncountably many curves called “rays”, each connecting ∞ to a well-defined “landing point” in C, so that every point in C is either on a unique ray or the landing point of finitely many rays. The key features of this paper are the following two: (1) this is the first example of a transcendental dynamical system where the Julia set is all of C and the dynamics is described in detail using symbolic dynamics; and (2) we get the strongest possible version (in the plane) of the “dimension paradox”: the set R of rays has Hausdorff dimension 1, and each point in C \ R is connected to ∞ by one or more disjoint rays in R; as a complement of a 1dimensional set, C \ R has of course Hausdorff dimension 2 and full Lebesgue measure.
منابع مشابه
Escaping Points and Symbolic Dynamics 4 3 Tails of Dynamic Rays 8 4 Dynamic Rays 14 5 Eventually Horizontal Escape 17 6 Classification of Escaping Points 21 7
We study the dynamics of iterated cosine maps E: z 7→ aez + be−z, with a, b ∈ C \ {0}. We show that the points which converge to ∞ under iteration are organized in the form of rays and, as in the exponential family, every escaping point is either on one of these rays or the landing point of a unique ray. Thus we get a complete classification of the escaping points of the cosine family, confirmi...
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